Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15);(1,3,6,8)(2,7,4,5)(10,12)(15,16)(17,18,23,24)(19,21,22,20);(2,4)(3,8)(9,11,14,13)(10,15,12,16)(17,19,23,22)(18,20,24,21);(3,8)(5,7)(9,10,14,12)(11,15,13,16)(17,20,23,21)(18,19,24,22);(1,4,6,2)(3,7,8,5)(9,12,14,10)(11,15,13,16)(17,21,23,20)(18,19,24,22);(1,5,6,7)(2,8,4,3)(9,13,14,11)(10,15,12,16)(17,22,23,19)(18,20,24,21)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=hi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72297120, #72297480 |
| maximal normal subgroup of | #72297120, #72297480 |
| maximal quotient of | #72297201, #72297254, #72297503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8564351);